This game differs from game 1 in that it has no dominant strategies. The rules are as follows: If player 1 plays a nickel, player 2 gives him 5 cents. If player 2 plays a nickel and player 1 plays a quarter, player 1 gets 25 cents. If both players play quarters, player 2 gets 25 cents. We get a payoff matrix for this game:
|Nickel  ||Quarter  |
|Player 1  ||Nickel||5||5|
Notice that there are no longer any dominant strategies. To solve this game, we need a more sophisticated approach. First, we can define lower and upper values of a game. These specify the least and most (on average) that a player can expect to win in the game if both player play rationally. To find the lower value of the game, first look at the minimum of the entries in each row. In our example, the first row has minimum value 5 and the second has minimum -25. The lower value of the game is the maximum of these numbers, or 5. In other words, player 1 expects to win at least an average of 5 cents per game. To find the upper value of the game, do the opposite. Look at the maximum of every column. In this case, these values are 25 and 5. The upper value of the game is the minimum of these numbers, or 5. So, on average, player 1 should win at most 5 cents per game.
|Nickel  ||Quarter  ||Min  |
Notice that, in our example, the upper and lower values of the game are the same. This is not always true; however, when it is, we just call this number the pure value of the game. The row with value 5 and the column with value 5 intersect in the top right entry of the payoff matrix. This entry is called the saddle point or minimax of the game and is both the smallest in its row and the largest in its column. The row and column that the saddle point belongs to are the best strategies for the players. So, in this example, player 1 should always play a nickel while player 2 should always play a quarter.